Derivatives

Study Material:

Introduction:

The derivative of a function of real variable measures the sensitivity to change of function value with respect to change of its argument.

For example, the derivative of the position of a moving

Point with respect to time is the velocity of that point.

Derivative can be generalized to functions of several variables.

The processing to find the derivative is called differentiation.

Definition:

Let be a real valued function and be any point on its domain. Then the derivative of at is denoted by and defined by

, provided this limit exists.

Or

, provided this limit exists.

To find the derivative of at any arbitrary point is given by

This definition of derivative is also called the first principle of derivative.

Remark:

There are different notations for derivative of a function. The derivative of , is denoted by or or D. Also the derivative of at the point is denoted by or

Example: Find the derivative of at arbitrary point

Solution:

.

Left hand derivative:

The left hand derivative of a real valued function at is denoted by and is defined by,

or

Right hand derivative:

The right hand derivative of a real valued function at is denoted by and is defined by,

or

Note:

The derivative of a function exists if and only if its left hand derivative and right hand derivative exist and equal that is exists if and only if .

Example: Find the left hand derivative and right hand derivative of the given function at

Solution:

And

.

Geometric representation of derivatives:

Let us consider a curve and draw a tangent of the curve at the point which makes an angle with the - axis.

Let be another point very close to the point. Add the points A and B.

Let

.

Then from the ABC,

Now, from the figure we say that when BA that is when the point B moves toward the point A, then unboundedly decrease and approach to and the line AB will approach to the tangent AP. Then

i.e.,

Algebra of derivative of functions:

Let and be two functions whose derivatives are defined in a common domain. Then

1. Derivative of sum of two functions is the sum of the derivatives of that two functions. That is

2. Derivative of difference of two functions is the difference of the derivatives of that two functions. That is

3. Derivative of product of two functions is the product of the derivatives of that two functions. That is

4. Derivative of quotient of two functions is the quotient of the derivatives of that two functions (whenever the denominator is non-zero). That is

.

Note:

In particular if we take, where is a constant then the formula 3 formed as

Remark: if and , then the above equations can be written as

1.

2.

3.

4. Respectively.

Basic formulas of derivatives:

General derivative formulas:

1. , where is any constant.

2. , where is positive or negative integer. This is called power rule of derivatives. In particular if we take , then .

3. , where is any function. This is called the power rule for functions.

Derivative of logarithm functions:

1.

2.

3.

4.

Derivative of exponential functions:

1.

2.

3.

4.

5.

Derivative of trigonometric functions:

1.

2.

3.

4.

5.

6.

Derivative of hyperbolic functions:

1.

2.

3.

4.

5.

6.

Derivative of inverse trigonometric functions:

1.

2.

3.

4.

5.

6.

Derivative of hyperbolic functions:

1.

2.

3.

4.

5.

6. .

Example 1: Prove that derivative of is .

Proof: Let then

.

(Proved).

Example 2: Compute the derivative of

Solution:

.

Solutions of the exercises of NCERT book

1. Find the derivative of at .

Solution:

Therefore the

.

2. Find the derivative of at .

Solution:

.

So the derivative of at is.

3. Find the derivative of at.

Solution:

.

4. Find the derivative of the following functions from first principle.

i).

Solution:

Let.

.

ii)

Solution:

Let .

.

iii).

Solution:

Let

.

iv)

Solution:

Let

.

5. For the function . Prove that .

Proof.

Here

So,

And .

Hence .

6. Find the derivative of , for some real number .

Solution:

.

7. For some constant and find the derivative of

(i)

Solution:

(ii)

Solution:

(iii)

Solution:

8. Find the derivative of for some constant.

Solution:

So,

9. Find the derivative of

(i)

Solution:

.

(ii)

Solution:

.

(iii)

Solution:

.

(iv)

Solution:

.

(v)

Solution:

.

(vi)

Solution:

10. Find the derivative of from first principle.

Solution:

Let then.

.

.

11. Find the derivative of the following functions:

(i)

Solution:

.

(ii)

Solution:

(iii)

Solution:

.

(iv)

Solution:

.

(v)

Solution:

.

(vi)

Solution:

.

(vii)

Solution:

.